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Lesson: Pascal's Triangle and the Binomial Theorem

Sample Question Videos

Worksheet • 4 Questions • 2 Videos

Q1:

Adam has been exploring the relationship between Pascal’s triangle and the binomial expansion. He has noticed that each row of Pascal’s triangle can be used to determine the coefficients of the binomial expansion of , as shown in the figure. For example, the fifth row of Pascal’s triangle can be used to determine the coefficients of the expansion of .

By calculating the next row of Pascal’s triangle, find the coefficients of the expansion of .

  • A1, 6, 15, 20, 15, 6, 1
  • B2, 8, 12, 8, 2
  • C1, 7, 21, 35, 35, 21, 7, 1
  • D1, 7, 21, 34, 34, 21, 7, 1
  • E1, 6, 8, 20, 8, 1

Adam now wants to calculate the coefficients for each of the terms of the expansion . By substituting into the expression above, or otherwise, calculate all of the coefficients of the expansion.

  • A16, 32, 24, 8, 1
  • B8, 32, 24, 8, 1
  • C8, 32, 24, 8, 1
  • D8, 64, 24, 8, 1
  • E16, 64, 24, 8, 1

Q2:

Find the coefficient of 𝑎 5 in the expansion of 𝑎 + 1 𝑎 𝑎 + 1 𝑎 2 2 3 3 .

Q3:

Mark knows that he can use the 6th row of Pascal’s triangle to calculate the coefficients of the expansion ( 𝑎 + 𝑏 ) 5 .

Calculate the numbers in the 6th row of Pascal’s triangle and, hence, write out the coefficients of the expansion ( 𝑎 + 𝑏 ) 5 .

  • A1, 5, 10, 10, 5, 1
  • B2, 6, 15, 20, 15, 6, 2
  • C1, 6, 15, 20, 15, 6, 1
  • D1, 3, 3, 1
  • E1, 7, 21, 35, 35, 21, 7, 1

Now, by considering the different powers of 𝑎 and 𝑏 and using Pascal’s triangle, work out the coefficients of the expansion ( 2 𝑎 2 𝑏 ) 5 .

  • A32, 160, 320, 320, 160, 32
  • B32, 160, 640, 640, 160, 32
  • C64, 160, 320, 320, 160, 64
  • D64, 160, 320, 640, 160, 64
  • E2, 10, 20, 20, 10, 2

Q4:

Shown is a partially filled-in picture of Pascal’s triangle. By spotting patterns, or otherwise, find the values of 𝑎 , 𝑏 , 𝑐 , and 𝑑 .

  • A 𝑎 = 1 0 , 𝑏 = 1 5 , 𝑐 = 2 1 , and 𝑑 = 1 1
  • B 𝑎 = 9 , 𝑏 = 1 2 , 𝑐 = 2 1 , and 𝑑 = 1 0
  • C 𝑎 = 6 , 𝑏 = 1 1 , 𝑐 = 2 1 , and 𝑑 = 1 0
  • D 𝑎 = 6 , 𝑏 = 1 1 , 𝑐 = 1 5 , and 𝑑 = 1 0
  • E 𝑎 = 1 0 , 𝑏 = 1 5 , 𝑐 = 2 0 , and 𝑑 = 1 1
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